ILL-POSEDNESS OF THE BASIC EQUATIONS OF FLUID DYNAMICS IN BESOV SPACES

We give a construction of a divergence-free vector field u(0) is an element of H(s) boolean AND B(infinity,infinity)(-1) for all s < 1/2, with arbitrarily small norm parallel to u(0)parallel to B(infinity,infinity)(-1) such that any any Leray-Hopf solution to the Navier-Stokes equation starting from u(0) is discontinuous at t = 0 in the metric of B(infinity,infinity)(-1). For the Euler equation a similar result is proved in all Besov spaces B(T,infinity)(S) where s > 0 if r > 2, and s > n(2/r - 1) if 1 <= r <= 2. This includes the space B(3,infinity)(1/3), which is known to be critical for the energy conservation in ideal fluids.